If $ f(x)=\log \frac{1+x}{1-x} $, then $ f(x) $ is

$ f\left(x_{1}\right) f\left(x_{2}\right)=f\left(x_{1}+x_{2}\right) $

$ \frac{f\left(x_{1}\right)}{f\left(x_{2}\right)}=f\left(x_{1}-x_{2}\right) $

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JEE • MATHEMATICS • FUNCTIONS

Functions practice · Easy

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Practice format and chapter quiz

These sets use single-correct-answer MCQs. They do not cover every JEE Main or JEE Advanced question format and are not full-length mock papers.

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Showing 331-360 of 478 questions

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331 Easy

331 of 478

Application +4 | -1

\phi(x)=x^2+1

and

\psi(x)=3^x

, then

\phi\{\psi(x)\}

and

\psi\{\phi(x)\}=

View answer and explanation

Correct option: C

3^{2 x}+1,3^{x^2+1}

Option C (

3^{2x} + 1, 3^{x^2+1}

) is correct because substituting

\psi(x)

into

\phi(x)

gives

3^{2x} + 1

and substituting

\phi(x)

into

\psi(x)

gives

3^{x^2+1}

332 Easy

332 of 478

Analyzing +4 | -1

f(x)=\left|\begin{array}{ccc}1 & x & (x+1) \\ 2 x & x(x-1) & (x+1) x \\ 3 x(x-1) & x(x-1)(x-2) & x(x-1)(x+1)\end{array}\right|

then

f(50)+f(51)+\ldots \ldots+f(99)

is equal to

View answer and explanation

Correct option: A

Option A (0) is correct because after applying the column operation C3 -> C3 - (C1 + C2), the determinant becomes zero for all x.

333 Easy

333 of 478

Application +4 | -1

f:R \to R\;and\;g:R \to R

are defined by

f\left( x \right) = 2x + 3\;and\;g\left( x \right) = {x^2} + 7,

then the values of x such that

g\left( {f\left( x \right)} \right) = 8

are

View answer and explanation

Correct option: C

- 1, - 2

Option C (-1, -2) is correct because these values satisfy the equation g(f(x)) = 8.

334 Easy

334 of 478

Analyzing +4 | -1

Let

f\left( x \right) = \left| {x - 1} \right|

Then,

View answer and explanation

Correct option: D

None of these

Option D None of these is correct because none of the given statements hold true for the function.

335 Easy

335 of 478

Application +4 | -1

Let

f

and

g

be two functions defined by

f(x)=\frac{x}{x+1}

g(x)=\frac{x}{1-x}

. Then

(f \circ g)^{-1}(x)

is equal to

View answer and explanation

Correct option: A

x

Option A (x) is correct because the composite function f(g(x)) simplifies to x, making the inverse function also x.

336 Easy

336 of 478

Application +4 | -1

Let

f:\left( { - \infty ,2} \right] \to \left( { - \infty ,4} \right]

be a function defined by

f\left( x \right) = 4x - {x^2}.

Then,

{f^{ - 1}}\left( x \right)

View answer and explanation

Correct option: A

2 - \sqrt {4 - x}

Option B (

\left[ { - 1 - \sqrt 3 \le x \le - 1 + \sqrt 3 } \right]

) is correct because it represents the values of x for which the expression inside the square root is non-negative.

337 Easy

337 of 478

Application +4 | -1

e^x=y+\sqrt{1+y^2}

, then

y=

View answer and explanation

Correct option: B

\frac{e^x-e^{-x}}{2}

Option B (

\frac{{e^x-e^{-x}}}{{2}}

) is correct because solving the given equation for

y

involves squaring and rearranging terms to isolate

y

338 Easy

338 of 478

Application +4 | -1

f(x)=\frac{1}{\sqrt{x+2 \sqrt{2 x-4}}}+\frac{1}{\sqrt{x-2 \sqrt{2 x-4}}}

for

x>2

, then

f(11)=

View answer and explanation

Correct option: C

6 / 7

Option C (

6 / 7

) is correct because simplifying the expression and substituting x=11 leads to the value 6/7.

339 Easy

339 of 478

Understanding +4 | -1

f(x)=\log \frac{1+x}{1-x}

, then

f(x)

View answer and explanation

Correct option: D

Odd function

Option D (Odd function) is correct because

f(-x) = -f(x)

, which is the definition of an odd function.

340 Easy

340 of 478

Understanding +4 | -1

The function

f(x)=\log \left(x+\sqrt{x^{2}+1}\right)

, is

View answer and explanation

Correct option: B

An odd function

Option B (An odd function) is correct because

f(-x) = -f(x)

, satisfying the definition of an odd function.

341 Easy

341 of 478

Application +4 | -1

If the functions

f

and

g

are defined by

f\left( x \right) = 3x - 4,\;g\left( x \right) = 3x + 2

, respectively then

{g^{ - 1}}\left( {{f^{ - 1}}\left( 5 \right)} \right) =

View answer and explanation

Correct option: C

1/3

Option C (1/3) is correct because it is the result of evaluating the composite inverse function.

342 Easy

342 of 478

Application +4 | -1

f(x+y)=f(x y) \, \forall \, x, y \in \mathbf{R}

, and

f(2013)

=2013

, then

f(-2013)

equals

View answer and explanation

Correct option: A

2013

Option A (2013) is correct because f(x+y) = f(xy) implies that f is a constant function, and since f(2013) = 2013, then f(-2013) must also be 2013.

343 Easy

343 of 478

Application +4 | -1

If the number of elements in

(A-B)-C,(B-C)

\sim A,(C-A)-B

and

A \cap B \cap C

10,15,20

, and 5 respectively then the number of elements in

(A \Delta B) \Delta C

View answer and explanation

Correct option: B

Option B (50) is correct because (A Δ B) Δ C represents the elements that belong to exactly one or all three of the sets, which sums to 50.

344 Easy

344 of 478

Understanding +4 | -1

A=\{1,2,3,4\}, B=\{1,2,3,4,5,6\}

are two sets and function

f: A \rightarrow B

is defined by

f(x)=x+2 ; \forall x \in A

, then the function

f

View answer and explanation

Correct option: C

One-one

Option C (One-one) is correct because each element in set A maps to a unique element in set B, satisfying the condition for an injective function.

345 Easy

345 of 478

Application +4 | -1

f(x+1)+f(x-1)=2 f(x)

and

f(0)=0

, then

f(n)

n \in N

, is

View answer and explanation

Correct option: A

n f(1)

Option A (

n f(1)

) is correct because the function follows a linear progression based on the given recurrence relation and initial condition.

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346 Easy

346 of 478

Analyzing +4 | -1

Let

f(x)=\sin \sqrt{x}

, then

View answer and explanation

Correct option: D

None of these

Option D (None of these) is correct because

f(x) = \sin \sqrt{x}

does not satisfy the periodicity condition

f(x+T) = f(x)

for any constant

T

347 Easy

347 of 478

Analyzing +4 | -1

Which of the following functions is (are) not an injective map(s)?

View answer and explanation

Correct option: B

\mathrm{g}\left( x \right) = x + \frac{1}{x},x \in \left( {0,\infty } \right)

The correct answer is

\mathrm{g}\left( x \right) = x + \frac{{1}}{{x}},x \in \left( {0,\infty } \right)

because it is not injective; g(2) = g(1/2) = 5/2, demonstrating that different inputs can yield the same output.

348 Easy

348 of 478

Application +4 | -1

2 f(x)+3 f(1 / x)=x^2-1

then

f(x)

View answer and explanation

Correct option: B

an even function

Option B is correct because the function f(x) satisfies the condition f(x) = f(-x), indicating it is an even function.

349 Easy

349 of 478

Understanding +4 | -1

Let

f: R \rightarrow R

be defined as

f(x)=3 x

. Choose the correct answer.

View answer and explanation

Correct option: A

f

is one-one onto

Option A (

f

is one-one onto) is correct because the function

f(x) = 3x

is both injective (one-to-one) and surjective (onto) over the real numbers.

350 Easy

350 of 478

Application +4 | -1

The range of

f(x)=\frac{x}{1+x^2}

is given by

View answer and explanation

Correct option: D

\left[-\frac{1}{2}, \frac{1}{2}\right]

Option D (

\left[-\frac{1}{2}, \frac{1}{2}\right]

) is correct because by setting

f(x) = y

and analyzing the discriminant of the resulting quadratic equation, we find the range of

y

to be between -1/2 and 1/2.

351 Easy

351 of 478

Understanding +4 | -1

The range of the function

f(x)=\frac{x-2}{2-x}

when

x \neq 2

View answer and explanation

Correct option: C

\{-1\}

Option C (

\{-1\}

) is correct because the function simplifies to -1 for all x not equal to 2.

352 Easy

352 of 478

Application +4 | -1

If the function

f:N \to N

is defined by

f\left( x \right) = \sqrt x

, then

\frac{{f\left( {25} \right)}}{{f\left( {16} \right) + f\left( 1 \right)}}

is equal to

View answer and explanation

Correct option: D

1

The correct answer is D (1) because substituting the values into the function and simplifying results in 1.

353 Easy

353 of 478

Application +4 | -1

\mathrm{f}:\left[\frac{\pi}{2}, \frac{3 \pi}{2}\right] \rightarrow[-1,1]

is defined by

f(x)=\sin x

, then

f^{-1}(x)

is given by

View answer and explanation

Correct option: C

\pi-\sin ^{-1} \mathrm{x}

The correct answer is C (

\pi-\sin ^{-1} \mathrm{x}

) because the inverse function is found by solving for x in terms of y and considering the restricted domain of the original function.

354 Easy

354 of 478

Application +4 | -1

f:\left[ {2,\;3} \right] \to R

is defined by

f\left( x \right) = {x^3} + 3x - 2

, then the range

f\left( x \right)

is contained in the interval

View answer and explanation

Correct option: B

\left[ {12,\mathrm{ }34} \right]

Option B ([12, 34]) is correct because the range is determined by evaluating the function at the endpoints of the interval [2, 3] since the function is monotonically increasing.

355 Easy

355 of 478

Application +4 | -1

Let

f(x)=\frac{1-x}{1+x}

. Then

f o f(\cos x)

is equal to

View answer and explanation

Correct option: B

\cos x

Option B (cos x) is correct because applying the function f twice results in the original input, cos x.

356 Easy

356 of 478

Application +4 | -1

f\left( x \right) = 2{x^6} + 3{x^4} + 4{x^2}

, then

f'\left( x \right)

View answer and explanation

Correct option: B

An odd function

Option B (An odd function) is correct because the derivative of an even function is always an odd function.

357 Easy

357 of 478

Application +4 | -1

If the function

f(x)

on the domain

\left[\frac{1}{2}, \infty\right)

is defined by

f(x)=2^{x(x-1)}

then

f^{-1}(x)

equals:

View answer and explanation

Correct option: A

\frac{1}{2}\left(1+\sqrt{1+4 \log _2 x}\right)

Option A (

\frac{1}{2}\left(1+\sqrt{1+4 \log _2 x}\right)

) is correct because it is the correct inverse function obtained by solving for x in terms of y and considering the domain.

358 Easy

358 of 478

Analyzing +4 | -1

The domain of the function

f\left( x \right) = \frac{{{{\sin }^{ - 1}}\left( {x - 3} \right)}}{{\sqrt {9 - {x^2}} }},

View answer and explanation

Correct option: B

\left[ {2,\;3} \right)

Option B is correct because it satisfies both conditions: -1 <= x-3 <= 1 and 9 - x² > 0, considering the denominator cannot be zero.

359 Easy

359 of 478

Application +4 | -1

f\left( x \right)

is an even function and

f'\left( x \right)

exists, then

f'\left( e \right) + f'\left( { - e} \right)

View answer and explanation

Correct option: B

= \mathrm{ }0

Option B (= 0) is correct because if f(x) is an even function, its derivative f'(x) is an odd function. Therefore, f'(-e) = -f'(e), and their sum is zero.

360 Easy

360 of 478

Analyzing +4 | -1

The function

f:R \to R

given by

f\left( x \right) = {x^2} + x,

, is

View answer and explanation

Correct option: D

Many one and into

Option D (Many one and into) is correct because the function is a parabola and doesn't cover the entire range of real numbers, and multiple x values can map to the same y value.

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